This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyt+? for weakly-R0 base spaces. Further, metrizability oft+? is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) oft+? is given in this general setting, thus generalizing results of [3], [5] and [6]

NC DOCKS at The University of North Carolina at Pembroke

Zsilinszky, Laszlo

English

The University of North Carolina at Greensboro

011τττ+++111+++111( , τ ) ( )1( ) 1 ( )1 τττ1X C L XXC L X C L XDi Maio HoláX THolá LeviTXNOTE ON HIT-AND-MISS TOPOLOGIES0. Introduction.RMathematics Subject Classification.Key words and phrases.NOTE ON HIT-AND-MISS TOPOLOGIESRENDICONTI DEL CIRCOLO MATEMATICO DI PALERMOSerie II, Tomo XLIX (2000), pp. 371-380LÁSZLÓ ZSILINSZKYThis is a continuation of [19]. We characterize first and second countabilityof the general hit-and-miss hyperspace topology for weakly- base spaces.Further, metrizability of is characterized with no preliminary conditions onthe base space and the generating family of closed sets and a new proof onuniformizability (i.e. complete regularity) of is given in this general setting,thus generalizing results of [3], [5] and [6].1991 54B20, Secondary 54D15, 54E35.Hit-and-miss topologies, countability and separation axioms,metrizability, uniformizability.371Let be a topological space and be the nonempty closedsubsets of . Following [2], [3], [5], [16], [17], [19], [20], [21] we willcontinue to study hit-and-miss hyperspace topologies or -topologies on, where is a fixed subfamily of . Two of the most studiedhit-and-miss topologies are the Vietoris topology ([14], [13]) and the Felltopology ([7], [13], [17]). In a recent paper [5], and havefound necessary and sufficient conditions for first and second countability,respectively of the -topology , if is ; more on countability axiomsand quasi-uniformizability of was obtained by and in [9],where a characterization of metrizability of is also given for a basespace and containing the singletons. Moreover, in [3] (see also [2]),N⋂FF1111111111 11ccniinn BVFcff+++−+′′ ′+ +=−+− −−LÁSZLÓ ZSILINSZKY1. Notation and terminology.⊂ ¯= { ∈∩ 6=∅} = { ∈ ; ⊂ }⊂∪∈ ∈ ∈==∈ ∈ ∈⊂ ⊂ ⊂→∈⊂ ⊂⊂ ∈= { ; ∈ } ∈τ11ττ( , τ ) ( )( ),( )( ) 1( ) 1 16(1 ) 11 τ ( ) ( )6(1) , . . . , τ( , . . . , ) 1 ( )τ 1 ( )τ1( ) 1 6(1), < α < β <6(1)( , α ) ( , β ),( ) ,( )( ) ( ) ( ).Beer Tamaki XXX C L XK XX E X E E EE X E A C L XA E E A C L X A EC L Xhit-and-misstopology C L X B UB U U nU U C L XK XUrysohn familyA C L X B DB D D Af X f fDf D ff M M fm infimal value functional C L Xm A f x x A A C L X372and characterized unifomizability of for a Hausdorff andcontaining the singletons.It is the purpose of this paper to show that quite similar characterizationshold with no preliminary conditions (or with much less restrictive conditions)on or , respectively. This is achieved by applying techniques andnotions from [19] and a completely new approach is employed to characterizecomplete regularity of .Note that a characterization of normality of is not known exceptfor some special cases, like the Vietoris topology ([11], [18]) or the Felltopology ([10]); for some more general results on normality see [6].In the sequel will be a topological space and (resp.) will denote the nonempty closed (resp. nonempty closed compact)subsets of . If , then , int will stand for the closure,interior and complement of , respectively in . Put ;, . In what follows, will be a fixedbut arbitrary nonempty subfamily of and for any , denoteby the set of all finite unions of members of . The or- for has a base all sets of the formwhere , and (cf. [3], [17]); this basicelement will be denoted by (cf; [20]). If , weobtain the familiar Vietoris topology , if , the Fell topology.In accordance with [3], is said to be a providedwhenever and are disjoint, there existssuch that int . Denote by the class of all continuousfunctions : [0 1] such that whenever inf sup , thereexists with[0 ] [0 ]where stands for the preimage of [0 1]. For denoteby the on (cf. [3]) defined byinf for all⋃N1( )1+ ++ +=+EMMAEMMAU VUU VU VUV0010 111 11 1012. Main results.NOTE ON HIT-AND-MISS TOPOLOGIESc cC L Xnmn B m Dc cicjcn B m Dc cc cc cicjci icjcmiic cicjcj i( )1τ( ) ( ) ( ) ( ), 6(1) , . . . ,, . . . , τ ( , )( , . . . , ) ( , . . . , )( , . . . , ) ( , . . . , )( ( ), τ ) ( )∈∈ ∈ ⊂ { }∩ 6=∅⊂∩ ∩∈∈ ∈⊂⊂ ≤ ≤ ≤ ≤∩ ⊂ ∩= =∈ \⊂ \∪ ∈ \ ⊂≤ ≤ ≤ ≤ ∅6= ∩ \ ∩⊂ ∩ \ ∩ ∈ \∈ ⊂ ⊂≤ ≤ ≤ ≤∩ ⊂ ∩ ∩ 6=∅ ∩ 6=∅ ∈ ut∈X has property P A C L Xx A D D A x D Xweakly R X PX X R spaceXE X c hemicompactK X C L E K X C L ETLet X be weakly-R , B D and U U ,V V m n . Then the following are equivalent:(i) U U V V ;(ii) B D and for every j m there exists an i n suchthat U B V D .Proof U U V VA B D RC B DA C B Dj m i n U B V DA U B V D AA A B Dj m i nU B V D A V A U AIf C L X is first countable, then every A C L Xis separable.Proof373We will say that provided whenever andthere exists such that and (see [19]).is called - provided possesses property or equivalentlyprovided every nonempty difference of -open sets contains a nonemptyclosed subset of ([19]). Further, is an - if every open subsetof contains the closure of each of its points ([4]).We will say that is - if there exists an increasingsequence of members of which is cofinal in .Notions not defined in the paper are used in accordance with [12] (e.g. regulardoes not include ).First we need some auxiliary material:L 2.1.1 1. Denote and . Suppose(i) and choose an . If is nonempty, then by the weak-property we can find a nonempty closed set . This implies that, which contradicts (i), thus . Further, if there existsa 1 such that for each 1 , , then wecan find a nonempty closed , but then ,which is a contradiction again, so (ii) holds.Conversely, suppose (ii) and pick an . Then .Further, for every 1 there is an 1 such that, so since . It means that .L 2.2.. The proof of Lemma 5.3 in [5] works in every topological spaceNN111BUU B0011101 1HEOREMHEOREM++++ −+′′′ ′+ ′ ′LÁSZLÓ ZSILINSZKYAcAxx x x n B xn x xn Bxcicicx xcn B nut⊂⊂ ∈⊂ ⊂ ∈⇒∈ = { }⊂ ⊂ = { ∈∈ ∈ }= ⊂∈ ∩ ⊂ ≤ ≤∈ ∩ { ∩ ; ∈ ∈ }ut⊂∈ ∈ ⊂ ⊂∈⊂ ⊂{ ; ∈ ∈ ∈ }( ( ), τ )1 1 16(1 )1 1 τ τ ( , . . . , ) 6(1 ), . . . , τ , ττ ( , . . . , )6(1 ), τ( ( ), τ )1 11 ( )6(1 )1 1 τ τ( , . . . , ) 6(1 ), , . . . , τ ,R XLet X be a weakly-R space. Then the following areequivalent:(i) Cl X is first countable;(ii) X is first countable, every closed set A X is separable and thereexists a countable family such that whenever B is disjointto A, then B D A for some D .ProofXx X A xU U BU U n AU x U U UB U U i nx B U B U B UxLet X be a weakly-R space. Then the following areequivalent:(i) C L X is second countable;(ii) X is second countable and there is a countable family suchthat whenever B and A C L X are disjoint, then B D Afor some D .ProofU U B U U n374if point-closures are used instead of singletons.We can now characterize first countability of the hit-and-miss topologyfor a weakly- base space (cf. [5], Theorem 5.4):T 2.3.. The proof of Theorem 5.4 in [5] can be abopted if Lemma2.1, Lemma 2.2 and point-closures are used insetad of singletons. In theimplication (i) (ii) only the proof of first countability of needs somecomments. Let and put . In view of (i) there exist countablefamilies and such that ; ,forms a countable local base at in . Chooseany -open neighborhood of . Then forsome , thus by Lemma 2.1, for an 1 andclearly . It means that is acountable local base at .As for second countability of the hit-and-miss topology we have (cf.[5], Theorem 5.13):T 2.4.. From (i) we get countable families , such thatNFFFFUEMMAHEOREMcfffcc cfn Bn31 111 11 12311111111111111+ ′ ′+− −−− −+ − − −++++++NOTE ON HIT-AND-MISS TOPOLOGIES{ ∩ ; ∈ ∈ }ut→∈∈ ∈{ ; ∈ } ∩ 6=∅− + ⊂ ∈∈+ ⊂ ⊂ +⊂ ∈ ∩ ⊂ ut⇒∈ ∈ ∈= = ∈ =∈ ∈ ∈τ 6(1 ), τ( ) , τ< α < β <((α, β)) α < ( ) < β ((α, β))< ε < ( ) α ( , α ε )6(1)( , α ε/ ) ( , α ε ),( ) ( ((α, β))) ((α, β))1( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )11 ( )( ) ( ) ( ) ( , . . . , )τ 6(1) , . . . , τB U B UXTThe functional m C L X is -continuousfor all f .Proof f f f Em f x x E E fm E f E fDf D fE D E D f mTThe following are equivalent(i) C L X is a Tychonoff space;(ii) C L X is completely regular;iii) C L X is a T -space;(iv) C L X is regular;(v) X has property P and is a Urysohn family.ProofD A C L X ff D f A A C L X U UA B U U n375forms a countable base of . Then is acountable base for , which easily follows by Lemma 2.1. The rest of theproof is analoguous to that of Theorem 5.13 in [5].It is shown in [19] that regularity and -ness of the Vietoris topologyare equivalent. We show that it is a general feature of hit-and-miss topologies.First we need the following:L 2.5. : [0 1]. Choose . Let inf sup and. Then inf , thus andfor any 0 we have [0 ] . Sincewe can find a such that[0 2] [0 ]whence . Then .The following theorem is proved in [3] (Theorem 3.6) for a basespace and with containing the singletons. Here we present a differentproof in the completely general setting:T 2.6.. (v) (i) In view of Theorem 1 in [19] it suffices to prove that thehyperspace is completely regular. An argument similar to that of in Lemma3.1 of [3] yields for all and disjoint an suchthat 0 and 1. Let and bea -neighborhood of , where , and .FUnnni0 10 000 1+++++++′′111111111111OROLLARYHEOREMcini i icif f ff f ff f fcif fcLÁSZLÓ ZSILINSZKY0 10 10 10⊂ ∩ 6=∅ ≤ ≤∈= { } = { }={ } = { } ≤ ≤= { − }− = = · · · = = =∈ ∩ 6=∅ ⊂ ≤ ≤− = = ==ut⊂∈ ∈∈ ⊂ ⊂ ⊂, , . . . ,( ) ( ) ,( ) ( ) ., , . . . , τ ( ), , . . . , τ( ) ( ) ( ) ( )/( ) ( ) ( )( )( ( ), τ )1( ( ), τ )1( ( ), τ )( ( ), τ )( ( ), τ )1 11 ( )6(1 )A B A U i nf f ff B f Af E f U i nm m m C L XF m m mm A m A m A F AE E B E U i nm E F E m EF EX PC L X RT TX XPC L X is uniformizable if and only if Xpossesses property P and is a Urysohn family.The following are equivalent:(i) C L X is metrizable;(ii) C L X is pseudo-metrizable;(iii) C L X is second countable and regular,(iv) X possesses property P and there exists a countable familysuch that whenever B and A C L X are disjoint there is aD with B int D D A .376Then and for all 1 . In virtue of the precedingconsiderations there exist functions such that0 and 10 and 1 for each 1The by Lemma 2.5, are -continuous onso max 1 is -continuous as well. Clearly1 0 so 0. Further ifthen either or for some 1 . In the firstcase 1 1, whence 1 and in the second case 1,so 1 again.All the remaining implications follows from Theorem 3 in [19], ifregularity of the hyperspace forces to have property . Indeed, ifis regular then it is also , further the hit-and-miss topology isalways (see [16]) so it is a -space (cf. [4], Corollary), which completesthe proof by Theorem 1 in [19].If is a Hausdorff space and contains the singletons then clearlypossesses property . Thus the following corollary generalises Theorem3.6 of [3]:C 2.7.Finally we turn to characterizing metrizability of the hit-and-misstopology:T 2.8.BVF1111++′′+03000HEOREMHEOREM3. Applications.NOTE ON HIT-AND-MISS TOPOLOGIES⇔⇒ ⇒⇒= { ; ∈ }∈ ∈⊂ ∩ { }6=∅∈ ⊂ ⊂ ⊂6= ∈ ⊂ ut⇒( ( ), τ )( ( ), τ )16(1 ) ( , τ )τ 16(1 )( ( ), τ )( ( ), τ )( ( ), τ )ProofC L X X RC L X T XR Xint D D X Ux U P BB U B xD B int D D UU X x D URemarkC L X XLet X be weakly-R . Then thefollowing are equivalent:(i) C L X is first countable;(ii) every closed subset of X is separable and has a countable base ofneighborhoods.Let X be weakly-R . Then thefollowing are equivalent:(i) C L X s first countable;(ii) X is first countable every closed set is separable and every properopen subset is c-hemicompact.Proof377. The equivalence (i) (ii) follows from Theorem 2.8. For(i) (iii) see Proposition 5.18(1) (2) in [5], further our Lemma 2.2 anduse point-closures instead of singletons. Now suppose (iii). Regularity ofimplies by Lemma 2(ii) of [19] that is weakly- , so ourTheorem 2.4 and Theorem 2.6 implies (iv) similarly as in [5] (Theorem5.19 (1) (2)). Finally, if we assume (iv) then according to Theorem 2.6,is a -space, consequently by Lemma 2(ii) of [19], isweakly- so if was second countable then in view of Theorem 2.4the -topology would be second countable and the Urysohn MetrizationTheorem would yield (i). Hence, it remains to justify that the countablefamily is a base for . Indeed, if is anonempty -open set and , then by property there existswith and . In virtue of the second condition of (iv) wecan find such that (we can assume that). Then int . .2.9. It is inferable from the proof of the preceding theoremthat metrizability of always forces second countability on .In view of our preceding theorems we have:T 3.1. (cf. [5], Theorem 5.5)T 3.2. (cf. [5], Theorem 5.6). (i) (ii) The proof of hemicompactness of proper open subsets30⇒ ⇒utut⇒ ˜˜ ˜˜ ˜VVVVVFFFFVVVV VLÁSZLÓ ZSILINSZKYHEOREMHEOREMHEOREM( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ )( ( ), τ ) τ ( )XcThe following are equivalent:(i) C L X is a Tychonoff space;(ii) C L X is completely regular;(iii) C L X is a T -space;(iv) C L X is regular;(v) C L X is uniformizable;(vi) X is normal and R .The following are equivalent:(i) C L X is a Tychonoff space;(ii) C L X is regular;(iii) C L X is a Hausdorff space;(iv) C L X is uniformizable;(v) X is a locally compact, regular space.ProofThe following are equivalent:(i) C L X is metrizable;(ii) X is compact and pseudo-metrizable.Proof C L X XXX C L XC L X C L X378of in [1], Lemma 3.1 is feasible also in our case if closed compact sets areused instead of compact sets and point-closures instead of singletons. Furthersee our Theorem 2.3. In (ii) (i) the proof of [5], Theorem 5.6 (2) (1)is applicable (using -hemicompactness instead of hemicompactness) alongwith our Theorem 2.3.T 3.3. (cf. [19]; Theorem 4)T 3.4.. Cf. [17] (Folgerung (a), p. 162) and Theorem 2 of [19].T 3.5. (cf. [14], Theorem 4.9.7). (i) (ii) Suppose that is metrizable. Denote bythe quotient space of induced by identification of points with commonclosure in . Then in view of Theorem 3 in [15], is homeo-morphic to , where is the Vietoris topology on ,( )( )0100VC L XFK X11226862˜ ˜⇒= ut⇒⇒=ut( ( ), τ )1 ( )( ( ), τ )( )( )( )1 ( )NOTE ON HIT-AND-MISS TOPOLOGIESC L XX RX T XX XXRP R XXC L XThe following are equivalent:(i) C L X is metrizable;(ii) X is locally compact, regular and second countable.Proof XXX K XXX K XK XK XX POn the Fell Topology,Topologies on Closed and Closed Convex Sets,The Infimal Value Functional and the Uniformization of Hit-and-Miss Hyperspace Topologies,Indexed systems of neighborhoods for general topological spaces,On hit-and-miss topologies,HEOREMREFERENCES[1] Beer G., Set-Valued Analysis, (1993), 69-80.[2] Beer G., Kluwer, Dordrecht, 1993.[3] Beer G., Tamaki R.,Proc. Amer. Math. Soc., (1994), 601-612.[4] Davis A., Amer.Math. Monthly, (1961), 886-893.[5] Di Maio G., Holá L’., Rend. Acc. Sc. Fis. Mat.Napoli, (1995), 103-124.379consequently it is also metrizable. Further, is a Hausdorff spa-ce so by Theorem 2 of [19], is regular and hence as well. Accordinglyis a -space, thus by Theorem 4.9.7 of [14], is compact, which im-plies compactness of (cf. [15], Theorem 4). Now is second countableby Remark 2.9, further it is regular, thus is pseudo-metrizable (see [8],p. 167, Exercise 3).(ii) (i) Observe that a pseudo-metrizable space is , hence possessesproperty (i.e. weak -ness). Further by Lemma 2.2, is a separable(pseudo-metrizable) space, accordingly second countable as well, whichtogether with compactness and regularity of easily yields the secondcondition of Theorem 2.8 (iv) for .T 3.6. (cf. [1], Theorem 3.4). (i) (ii) According to Remark 2.9, is second countable andin virtue of Theorem 3.4, is locally compact and regular.(ii) (i) By local compactness plus regularity of , forms a baseof neighborhoods for closed compact subsets of ([12], p. 146, Theorem18). Further, second countability of yields a countable subfamily ofwhich forms also a base of neighboroods for members of , thus thesecond condition of Theorem 2.8 (iv) is fulfilled for . Finally,local compactness and regularity of evidently imply property , thusTheorem 2.8 applies.X0380135127337134165916164512449LÁSZLÓ ZSILINSZKYPervenuto il 10 luglio 1996,In forma modificata il 2 luglio 1999.Notes on hit-and-miss topologies,A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space,Point Set Topology,Decomposition properties of hyperspaces topologies,Normality and Paracompactness of the Fell topology,On the equivalence of normality and compactness in hyperspaces,General Topology,Theory of Correspondences,Topologies on spaces of subsets,Some proeprties of the space of a topological R -space,Eine Bemerkung über Trennungsaxiome in Räumen von abgeschlosse-nen Teilmengen topologisher Räume,Einige Bemerkungen über den Raum der abgeschlossenen Mengen,On the space of closed subsets,On separation axioms in hyperspaces,Baire space and hyperspace topologies,Baire spaces and weak topologies generated by gap and excessfunctionals,Department of Mathematicsand Computer ScienceUniversity of North Carolinaat PembrokePembroke, NC 28372 - USAE-mail address: laszlo@nat.uncp.edu[6] Di Maio G., Holá L’., Meccariello E., RostockerMath. Kolloq., to appear.[7] Fell J.,Proc. Amer. Math. Soc., (1962), 472-476.[8] Gaal S. A., Academic Press, New York, 1964.[9] Holá L’., Levi S., Set-ValuedAnal., (1997), 309-321.[10] Holá L’., Levi S., Pelant J.,Proc. Amer. Math., Soc., (1999), 2193-2197.[11] Keesling J.,Pacific J. Math., (1970), 657-667.[12] Kelley J. L., Springer-Verlag, New York, 1975.[13] Klein E., Thompson A., Wiley, New York, 1975.[14] Michael E., Trans. Amer. Math. Soc., (1951),152-182.[15] Mrševič M., 2 UspechiMat. Nauk, n. 6 (210) (1979), 166-170 (Russian).[16] Poppe H.,Arch. Math., (1965), 197-199.[17] Poppe H.,Fund. Math., (1966), 159-169.[18] Veličko N. H., Sibirsk. Math. Z. (1975),627-629. (Russian; English translation: Siberian Math. J., (1975), 484-486).[19] Zsilinszky L., Rend. Circ. Matem. Palermo,(1996), 75-83.[20] Zsilinszky L., Proc. Amer. Math. Soc.,(1996), 3175-3184.[21] Zsilinszky L.,Math. Slovaca, (1999), 357-366.

Note on Hit-And-Miss Topologies

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Note on Hit-And-Miss Topologies

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